Hadamard product nonlinear formulation of Galerkin and finite element methods

TL;DR

This paper introduces a new nonlinear formulation for finite element and Galerkin methods using Hadamard products, reducing computational costs and enabling nonlinear decoupling.

Contribution

It presents a novel Hadamard product-based nonlinear formulation that simplifies computations and avoids repeated numerical integration in finite element and Galerkin methods.

Findings

Reduces computational cost by avoiding repeated numerical integration.

Enables nonlinear decoupling computations.

Maintains advantages of weak formulation in standard schemes.

Abstract

A novel nonlinear formulation of the finite element and Galerkin methods is presented here, which leads to the Hadamard product expression of the resultant nonlinear algebraic analogue. The presented formulation attains the advantages of weak formulation in the standard finite element and Galerkin schemes and avoids the costly repeated numerical integration of the Jacobian matrix via the recently developed SJT product approach. This also provides possibility of the nonlinear decoupling computations.